Gårding’s Inequality During Three Decades

Abstract

1. The original result. The Gårding inequality was first published in 1953 and was stated as follows in [4]: Theorem 1.1. Let p αβ (x) be real valued and symmetric in the multi-indices α, β with |α| = |β| = m, uniformly continuous in Ω ⊂ R n. If (1.1) p αβ (x)ξ α+β ≥ c|ξ| 2m when x ∈ Ω, ξ ∈ R n , then there is a constant K such that (1.2) (p αβ D α u, D β u) + K(u, u) ≥ 0, u ∈ C ∞ 0 (Ω). Here (,) is the scalar product in L 2 where the norm will be written. A stronger statement is given during the proof, (1.2) ′ u 2 (m) = |α|≤m D α u 2 ≤ C (p αβ D α u, D β u) + Ku 2 , u ∈ C ∞ 0 (Ω), and this stronger statement is essential in the proof. Indeed, in view of the interpolation inequality (1.3) u (k) ≤ Cu k/m (m) u (m−k)/m (0) the estimate (1.2) ′ follows by means of a partition of unity if it is known when u has small support. By the uniform continuity assumed it suffices therefore to prove (1.2) ′ with coefficients frozen at any point in Ω. However, in the constant coefficient case (1.2) ′ follows from Parseval's formula which allows one to exploit directly the positivity in (1.1). When m = 1 one can of course write p αβ (x)ξ α+β as a sum of squares to prove the estimate. However, this is in general impossible in the higher order case, so the use of Fourier analysis essential. This was one of the essential points in Gårding's paper; the other was the effective use of a partition of unity. The stability of (1.2) ′ under small perturbations shows that one may assume that p αβ and all their derivatives are defined and bounded in the whole of R n. Integrating by parts and using (1.4) we see that the result is then equivalent to (1.4) u 2 (m) ≤ C Re (P (x, D)u, u) + Ku 2 , u ∈ C ∞ 0 (R n), 16 Lecture at the Gårding symposium May 31, 1985. 137 L. Hörmander, Unpublished Manuscripts, https://doi.